A Multiplication Based Logic Puzzle

  • 373 is a prime number.
  • Prime factorization: 373 is prime.
  • The exponent of prime number 373 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 373 has exactly 2 factors.
  • Factors of 373: 1, 373
  • Factor pairs: 373 = 1 x 373
  • 373 has no square factors that allow its square root to be simplified. √373 ≈ 19.313

How do we know that 373 is a prime number? If 373 were not a prime number, then it would be divisible by at least one prime number less than or equal to √373 ≈ 19.313. Since 373 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 373 is a prime number.

  • 373 is the short leg of only one Pythagorean triple, the primitive 373, 69564, 69565
  • Prime numbers are never the longer leg, but
  • 373 is the hypotenuse of exactly one Pythagorean triple: 252, 275, 373

Here is the Odd Pythagorean triple sequence I’ve blogged about this week with the prime numbers highlighted in yellow:

Odd Primitive Primes

There are 24 odd prime numbers less than 100. The odd numbers less than 100 in this sequence produce 21 prime numbers as their hypotenuses! I think that is amazing especially since 40% of the time the hypotenuse turns out to be a composite number whose last digit is five! Here are some observations that apply to THIS sequence only:

  • When the last digit of the short leg is 3 or 7, the last digit of the hypotenuse ends in 5.
  • When the last digit of the short leg is 5, the last digit of the hypotenuse ends in 3.
  • When the last digit of the short leg is 1 or 9, the last digit of the hypotenuse ends in 1.

There are only 18 prime hypotenuses when we use about the same number of triples from this Even Primitive Triple Sequence.

Even Primitive Primes

18 primes out of 49 numbers listed is slightly less impressive than 21 primes out of 48 total numbers, but again 40% of the hypotenuses end with five and have no choice but to be composite numbers.  Here are some observations that apply to THIS sequence only:

  • When the last digit of the short leg is 4 or 6, the last digit of the hypotenuse ends in 5.
  • When the last digit of the short leg is 0, the last digit of the hypotenuse ends in 1.
  • When the last digit of the short leg is 2 or 8, the last digit of the hypotenuse ends in 7.

I had no idea that Pythagorean triples would produce so much trivia!

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